Three Separations, One Method: A Vertical Demonstration Across the Identity–Persistence Program

Abstract This paper develops three formal separation results within the Identity–Persistence Program while simultaneously demonstrating the program’s general proof method. The three results establish, by explicit construction, that identity is not recognition, identity is not admissibility, and persistence is not stability. Each distinction is proved using the same sequence of operations: declaring the level at which the distinction arises, identifying the forced structural role involved, constructing a separating model, determining the declaration required to legitimately collapse the distinction, and classifying the remaining uncertainty. The principal contribution is methodological rather than foundational. By executing the same procedure across three independent non-identities, the paper exposes a common proof strategy that has been used throughout the corpus but had not previously been stated explicitly. The resulting method provides a systematic procedure for converting informal conceptual distinctions into formally auditable constructions while preserving explicit status discipline. The paper introduces no new forcing theorem and makes no claim of methodological exhaustiveness. It serves as a methods-layer demonstration connecting the Orientation, Identity, Admissibility, Sufficient Regime Specification, Root of the Demand, Algebra, and Definability papers into a single reproducible workflow. Remaining theoretical and applied developments—including the Aboutness Lemma and the Minimal Decision Record—are identified but intentionally left to separate companion papers.